There are two types of security in Access; the database password, and
user-level security.
The database password prompts you only for a password, and is applied to the
mdb file. User-level security asks you for a username and password. Since
you are getting a login prompt no matter what mdb you open, it sounds like
you're talking about ULS.
Access uses user-level security out of the box. It ships with a workgroup
file called system.mdw and uses this for all sessions (it silently logs you
in as a user named 'Admin'). When you implement security, you create a new
mdw file and then set a password on the Admin user. Doing so causes the
login dialog to appear.
You are always joined by default to some mdw file. If a password has been
set for the Admin user in the mdw file you are joined to, then you'll get a
login prompt.
You want to rejoin system.mdw so that you don't get a login prompt. Use
search to locate all mdw files on your computer. Try joining each one (use
Tools, security workgroup administrator to join a workgroup as your
default). After joining one, try opening any mdb file. If you aren't
prompted for a username/password, then you have the right one. Now you
should be able to open any mdb file without a login.